For a \(d\) -dimensional system of linear size \(L\) , the characteristic frequency \(\omega _c\)  scales according to \(\omega _c(L) \sim L^{-z}\)  with \(z\)  the dynamical exponent. In the case of infinite-range systems, \(L\)  is replaced by \(N^{1/d}\)  with \(d\)  the upper critical dimension (\(d=4\)  in the present case) . In Fig. REF , the frequency is therefore rescaled to \(\omega _c(N) \sim N^{-z/d}\) , with the choice \(z=1\)  providing excellent collapse of the data for various system sizes. In summary, Fig. REF  shows that the dynamical exponent is \(z=1\)  at the thermal transition. Interestingly, the value \(z=1\)  differs from the mean-field prediction \(z=2\)  for the critical dynamics of a non-conserved order parameter (model A) [1]}, [2]}. However, \(z=1\)  is in agreement with both the quantum O(1) model in a gaussian approximation ; and with the quantum spherical model, which has the same symmetries as the Ising model, and which for \(x=2\)  has the same (mean-field) static critical exponents.
